#183499

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(6,11,13,17)(7,14)(12,19,20,18)(15,16)(21,22,25,30,26,28,24,29)(23,31,32,27,34,36,33,35);(1,2,3,4)(5,6)(7,12)(8,11)(9,17)(10,13)(14,20)(15,19)(16,18)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,36)(30,31);(5,7,9,15,10,14,8,16)(6,12,17,19,13,20,11,18)(22,28)(27,35)(29,30)(31,36);(5,8,10,9)(6,11,13,17)(7,15,14,16)(12,19,20,18)(21,24,26,25)(22,29,28,30)(23,32,34,33)(27,36,35,31)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,e^b=eh,f^b=fh,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90695);